Derivation of LLG equation in polar coordinates

In summary, the torque contribution due to uniaxial anisotropy can be represented by the equation (3) in cartesian coordinates. By using this contribution in the LLG equation (9) and converting it to polar coordinates, a differential equation in the two angles can be derived. The final result of this derivation is given in the form of a matrix (theta', phi') as shown in the paper. However, the process of deriving this result may require additional resources and may vary depending on the specific system being studied.
  • #1
apervaiz
2
0
The torque contribution due to the uniaxial anisotropy is given by the equation below

[tex] \frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3)[/tex]

This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates

[tex] \frac{\partial n_m}{\partial t} + ( n_m \times \frac{\partial n_m}{\partial t})=\frac{1}{2}\Omega_K \frac{\Gamma}{l_m K} (9) [/tex]
where [itex]n_m= [r,\theta,\phi][/itex]. Since r is unity for magnetization a differential equation in the two angles should be possible which should have the form
[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\theta \\
\phi \\
\end{bmatrix}
[/tex]

Now the result of this derivation is already given as

[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\alpha \sin \theta \cos \theta \\
\cos \theta \\
\end{bmatrix}
[/tex]

I'm having a hard time deriving this result from (3) using (9). Could anyone help me with this?
here is the link of this paper
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.570
 
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  • #2
Can you be more specific about where you are having a hard time? Is it the change of coordinate system?

Also, when asking a question like this, it's quite reasonable to mention that you're setting [itex]h_p=h=h_s=0[/itex]. Makes it easier to relate your equations to those in the paper.
 
  • #3
Thanks for the reply. You are right this equation only caters for uniaxial anisotropy. the equation has to be derived in polar angles while (3) is in cartesian. I think going from (3) to polar is not hard, but how to get a clean expression like the final one.
I'm not sure how to proceed for the derivation at all.
 
  • #4
Last edited by a moderator:

1. What is the LLG equation?

The LLG equation, also known as the Landau-Lifshitz-Gilbert equation, is a mathematical model that describes the dynamics of magnetization in ferromagnetic materials. It takes into account the effects of both precession and damping on the magnetization vector.

2. Why is the LLG equation important?

The LLG equation is important because it provides a fundamental understanding of how magnetization behaves in ferromagnetic materials. It is also the basis for many practical applications, such as magnetic data storage and spintronics.

3. What are polar coordinates?

Polar coordinates are a system of coordinates used to describe a point in a two-dimensional space. They consist of a radial distance from the origin and an angle measured from a fixed reference direction.

4. How is the LLG equation derived in polar coordinates?

The LLG equation can be derived in polar coordinates by first expressing the magnetization vector in terms of its polar components. Then, using the chain rule and some vector calculus, the equation can be transformed into its polar form.

5. What are the assumptions made in deriving the LLG equation in polar coordinates?

The derivation of the LLG equation in polar coordinates assumes a uniform magnetization and neglects any external magnetic fields. It also assumes that the damping coefficient is constant and that the magnetization is not affected by thermal fluctuations.

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